Phase transitions in the condition number distribution of Gaussian random matrices

We study the statistics of the condition number $\kappa ={\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\mathrm{/}{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ (the ratio between largest and smallest squared singular values) of $N\times M$ Gaussian random matrices. Using a Coulomb fluid technique, we derive analytically and for large $N$ the cumulative $\mathcal{P}[\kappa <x]$ and tail-cumulative $\mathcal{P}[\kappa >x]$ distributions of $\kappa$. We find that these distributions decay as $$\mathcal{P}$[\kappa<x]\approx\exp\left(-\beta N^2 \Phi_{-}(x)\right)$and$$\mathcal{P}$[\kappa>x]\approx\exp\left(-\beta N \Phi_{+}(x)\right)$,where$\beta$ is the Dyson index of the ensemble. The left and right rate functions ${\mathrm{\Phi }}_{\pm }(x)$ are independent of $\beta$ and calculated exactly for any choice of the rectangularity parameter $\alpha =M\mathrm{/}N-1>0$. Interestingly, they show a weak non-analytic behavior at their minimum $⟨\kappa ⟩$ (corresponding to the average condition number), a direct consequence of a phase transition in the associated Coulomb fluid problem. Matching the behavior of the rate functions around $⟨\kappa ⟩$, we determine exactly the scale of typical fluctuations $\sim \mathcal{O}(N^{-2\mathrm{/}3})$ and the tails of the limiting distribution of $\kappa$. The analytical results are in excellent agreement with numerical simulations.